Having received his early education at Peterhead Academy and Aberdeen Grammar School, he had a brilliant career at the University of Aberdeen, where he was awarded several prizes, a gold medal and the Fullerton and Ferguson Scholarships. He graduated with first-class honours in Mathematics and Natural Philosophy in 1903 and then went to Clare College, Cambridge. He became Fourth Wrangler in 1906, and obtained a first class in Part II of the Mathematical Tripos. He was mathematics master at Clifton College, Bristol, from 1907 until 1919, and in the latter year was appointed Professor of Mathematics at the University of Leeds, a post held by him until he retired in 1946. He went to Leeds as the sole member of the Department of Mathematics, receiving during his first term only the help of a graduate in engineering, but he was joined in December 1919, by C W Gilham, who in due course became a Senior Lecturer in Mathematics; and in the following month by S Brodetsky, who, after being Reader in Applied Mathematics, was made Professor of Applied Mathematics in 1924. The Milne-Brodetsky professorial partnership became famous in academic circles, not only because of their reputation as mathematicians, but also because they were both men of outstanding personality. Among other well-known mathematicians who were members of the staff at various times during Milne's headship of the mathematics department were W E H Berwick, R M Gabriel, A E Ingham, Glenny Smeal, R Stoneley and H D Ursell.
Milne was a man of lively disposition who on social occasions would often be the centre of a hilarious group of people. He showed great zest in all his widespread activities and much force of character in achieving his objectives. Of his many services to the University of Leeds and to university education generally, perhaps the greatest was that of acting as Pro-Vice-Chancellor during the second world war, when, in the absence abroad of the Vice-Chancellor, he gave compelling leadership to the committee concerned with the university's Post-War Development Plan. He was an enthusiastic supporter of the Mathematical Association, believing it to provide an essential link between mathematics teachers in schools, colleges and universities: in 1921 he persuaded the Association to hold in Leeds the first annual general meeting ever to take place outside London, and he was instrumental in founding the Yorkshire branch of the Association.
He had many interests, of which one needs special mention: he was fascinated by languages, and especially by Gaelic. The promotion of Celtic studies was indeed an interest of his barely second to that of mathematics. After his retirement he completed a task upon which he had been engaged for some years, unknown to anyone outside his family, namely the writing of a novel Eppie Elrick (An Aberdeenshire tale of the '15) [Published by Scrogie, Peterhead, 1956]. In it all the dialogue is in Scots, and it was intended among other things to record the dialect, as it existed in his youth, of his native Buchan district of Aberdeenshire. The book was such a success in Aberdeenshire that a reprint was required a few months after its publication.
In his early days as a schoolmaster Milne wrote, or collaborated in, five mathematical text-books [Homogeneous Coordinates for use in colleges and schools (1910), Projective Geometry for use in colleges and schools (1911), Higher Algebra (1913), (with G J B Westcott) A first course in Calculus Parts I, II (1919-20), (with G J B Westcott) The elements of the calculus (1927)] notable in their day for the clarity of their style, all of them running into several editions or reprints.
His great distinction as a man, as a mathematician and as a university statesman was recognized by the award of the honorary degree of LL.D. by the University of Aberdeen at the time of his retirement in 1946, and by the University of Leeds in 1955.
Among those who taught Milne in his undergraduate days was J H Grace: a colourful personality whose lectures, whether or not diversified by an occasional irregularity, are deemed to have been among the most brilliant and inspiring of that era. Grace was recognised on all sides as an authority on invariants, but he was also a geometer of striking originality and fertile imagination, and contact with him was a potent factor in determining the line that Milne would follow. And so, through Grace's help, a young schoolmaster at Clifton College was soon writing alongside H F Baker and G H Hardy in our Proceedings.
These early papers of Milne date from 1910 and handle problems posed by the apolarity of a plane cubic G and triads of its own points. ...
It is perhaps permissible, after the lapse of time, to disclose that the present writer [W L Edge] refereed A triad of quadrinodal cubic surfaces containing a quadricublic curve, Proc. London Math. Soc. 44 (1938), 345-351] in 1937; the paper still glows with the bracing zest and clarity that graced its birth 30 years ago.